Morphisms from a very general hypersurface

TL;DR

This paper investigates morphisms from very general hypersurfaces, establishing bounds on their degrees and characterizing the target varieties, including conditions under which the target is projective space.

Contribution

It provides new bounds on the degree of surjective morphisms from hypersurfaces and characterizes the target varieties, especially when they are factorial or smooth.

Findings

Y is a klt Fano variety if deg f is large enough

An optimal upper bound deg f ≤ deg X is established for factorial targets

Y is isomorphic to projective space under certain conditions

Abstract

Let $X$ be a very general hypersurface of degree $d$ in the projective $(n+1)$-space with $n \ge 3$, and $f: X \to Y$ a non-birational surjective morphism to a normal projective variety $Y$. We first prove that $Y$ is a klt Fano variety if ${\rm deg} \, f \ge C$ for some constant $C = C(n, d)$ depending only on $n$ and $d$. Next we prove an optimal upper bound ${\rm deg} \, f \le {\rm deg} \, X$ provided that $Y$ is factorial, ${\rm deg} \, f$ is prime and ${\rm deg} \, f \ge E(n)$ for some constant $E(n)$ (with $E(n) = n(n+1)$ when $Y$ is smooth). As a corollary, we show that $Y\cong {\bf P}^n$ under some conditions on $Y$ and ${\rm deg} \, f$.